Blackjack, Blinder, Bear, Flanker, Fulcrum, Fencer, Frogfoot, Comb, Condor, Cub, Hind, Halo, Hip, Hokum, etc etc are all NATO code/reporting names.
Alligator is Kamov's name for it's attack helo. It probably was given a name largely because it was intended at the outset to be an export product, and the export market is accustomed to have names assigned to products.
Tu-22 NATO code was Blinder. Soviet crews called it "shilo" which translates to "awl" because of its shape.
Perhaps one of the reasons for the lack of names is the somewhat unique design/development process used by the old Soviet system. A "design bureau" designs the basic product and (usually) an entirely different entity (sometimes more than one entity) builds and sustains it. The production entity has its own internal "design bureau" with the authority to make changes to the basic design to improve manufacturability or assembly, or to facilitate the incorporation of new material and/or processes as they became available and/or to improve the product itself. The production center has different motivations (and funding) than the design bureau and with the authority to make significant design changes, is why there are so many significantly different variants of Soviet/Russian aircraft.
For example, the Mi-8/17 helo is built in both Kazan and Ulan-Ude. Each production center competes with the other and is why each has its own quite different version(s) of the Mi-8/17.